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(cf. Fig.
2.8
b). The length of a perpendicular
k
depends on the local uncertainty
σ
k
of the curve given by
σ
k
n
k
J
C
k
·
=
·
J
C
k
·
Σ
T
·
n
k
(2.12)
and changes in each iteration step. The variable
n
k
defines the curve normal and
J
C
k
the Jacobian, i.e. the partial derivative of the curve with respect to the model
parameters
T
. The original real-time CCD algorithm according to Hanek (
2001
)
computes for every perpendicular
k
a local uncertainty
σ
k
of the curve. In contrast
to Hanek (
2001
), we apply a fixed local uncertainty for every perpendicular of the
curve, thus avoiding a large number of nonlinear function evaluations. The local
uncertainties
σ
0
and
σ
K/
2
at the perpendiculars
k
=
0 and
k
=
K/
2 are computed
using (
2.12
), and the fixed local uncertainty
σ
results from
σ
σ
K/
2
)/
2. The
fixed local uncertainty
σ
is now used to compute the probabilistic assignment for
every point
v
k,l
on the perpendiculars;
l
denotes the pixel index on perpendicular
k
.
For every point
v
k,l
on perpendicular
k
the probability
a
l,
1
(d
l
)
that the point lies on
side 1 (inner side) of the curve (probabilistic assignment) is computed by
=
(σ
0
+
erf
d
l
1
.
1
2
a
l,
1
(d
l
)
=
√
2
σ
+
(2.13)
n
k
(
v
k,l
−
In (
2.13
), erf
(x)
is the Gaussian error function and
d
l
(
v
k,l
)
C
k
)
the
signed distance of the pixel coordinate
v
k,l
from the curve point
C
k
. The probabilis-
tic assignment
a
l,
2
(d
l
)
of side 2 (outer side) is defined by
a
l,
2
(d
l
)
=
=
−
a
l,
1
(d
l
)
.
To compute the two-sided probability distributions of the pixel grey values we
follow the suggestions of Hanek (
2001
) and apply as a weighting function the ex-
pression
1
w
l,s
=
w
A
(a
l,s
)
·
w
B
(d
l
,σ)
·
w
C
(σ )
with
s
∈{
1
,
2
}
,
(2.14)
where
max
0
,
a
l,s
−
(
2
·
E
A
)
γ
1
w
A
(a
l,s
)
=
(2.15)
1
−
γ
1
assesses the probabilistic assignment. The parameter
γ
1
∈[
describes the mini-
mum probability
a
l,s
that the pixel is used to compute the probability distributions.
We use
γ
1
=
0
,
1
[
3. The weight
w
B
(d
l
,σ)
considers the signed distance
of the pixel
v
k,l
to the curve and is given by
0
.
5 and
E
A
=
max
0
,e
(
−
d
l
/(
2
·
σ))
e
(
−
γ
4
)
with
w
B
(d
l
,σ)
=
C
·
−
(2.16)
σ
=
γ
3
·
σ
+
γ
4
.
(2.17)
Here
C
is a normalisation constant, where we set
C
=
5. For the other parameters
we use
γ
2
=
4,
γ
3
=
6, and
γ
4
=
3. The weighting function
w
C
(σ )
evaluates the
local uncertainty
σ
according to
1
)
−
E
C
.
w
C
(σ )
=
(σ
+
(2.18)
We recommend the choice
E
C
=
4. Since we consider a fixed local uncertainty
σ
,
the probabilistic assignment and the weights depend only on the signed distance
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